The Shenandoah River (VA, USA), the largest tributary of the Potomac River (MD, USA) and an important source of drinking water, has been the site of extensive fish kills since 2004. Previous investigations indicate environmental stressors may be adversely modulating the immune system of smallmouth bass (Micropterus dolomieu) and other species. Anterior kidney (AK) tissue, the major site of bloo...

We show that for any given n, there exists a sequence of words {ak}k≥1 in the generators σ1, . . . , σn−1 of the braid group Bn, representing the identity element of Bn, such that the number of braid relations of the form σiσi+1σi = σi+1σiσi+1 needed to pass from ak to the empty word is quadratic with respect to the length of ak.

has density one in Z with respect to some sequence of intervals Ik = [ak, bk] with bk−ak → ∞. (This means that d{Ik}(S) = lim k→∞ |S∩Ik| bk−ak+1 = 1.) A vector f ∈ H is called weakly wandering if there is an infinite set S ⊆ Z such that for any n,m ∈ S, n 6= m, one has 〈Uf, Uf〉 = 0. The following theorem due to U. Krengel gives a characterization of weak mixing in terms of weakly wandering vect...

Abstract Let A1, . . . , Ak be n×n matrices. We studied inequalities and equalities involving eigenvalues, diagonal entries, and singular values of A0 = A1 · · ·Ak and those of A1, . . . , Ak. It is shown that the matrices attaining equalities often have special reducible structure. The results are then applied to study normality and reducibility of matrices, extending some results and answerin...